Toán Lớp 10: Cho các số thực $a,b,c$ thỏa mãn $(a^2+b^2)(b^2+c^2)(a^2+c^2)=8$ Chứng minh rằng: $(a^2-ab+b^2)(b^2-ab+c^2)(c^2-ac+a^2) ≥ 1$

Question

Toán Lớp 10:  Cho các số thực $a,b,c$ thỏa mãn $(a^2+b^2)(b^2+c^2)(a^2+c^2)=8$ Chứng minh rằng: $(a^2-ab+b^2)(b^2-ab+c^2)(c^2-ac+a^2) ≥ 1$

thanhtung · Answer

$ text{Ta sẽ chứng minh :2.(a&sup2;-ab+b&sup2;)&sup2;&ge; $a^{4}$ +$b^{4}$ }$    $ text{ &hArr;2[(a&sup2;+b&sup2;)-ab]&sup2;&ge; $a^{4}$ +$b^{4}$  }$   $ text{  &hArr;2.[ (a&sup2;+b&sup2;)&sup2;-2.ab(a&sup2;+b&sup2;)+a&sup2;.b&sup2;]&ge; $a^{4}$ +$b^{4}$ }$     &hArr;2.(a ⁴+2.a&sup2;.b&sup2;+b⁴-2ab(a&sup2;+b&sup2;)+a&sup2;b&sup2; )&ge;$a^{4}$ +$b^{4}$       $ text{&hArr;2.a⁴+4.a&sup2;b&sup2;+2.b⁴-4.ab(a&sup2;+b&sup2;)+2a&sup2;b&sup2;&ge;$a^{4}$ +$b^{4}$ }$      $ text{&hArr;a⁴+4.a&sup2;b&sup2;+b⁴-4.ab(a&sup2;+b&sup2;)+2a&sup2;b&sup2;&ge;0}$      $ text{&hArr;(a⁴+2a&sup2;b&sup2;+b⁴)-4.ab(a&sup2;+b&sup2;)+(2ab)&sup2;&ge;0}$      $ text{&hArr;(a&sup2;+b&sup2;)&sup2;-4.ab(a&sup2;+b&sup2;)+(2ab)&sup2;&ge;0}$      $ text{&hArr;(a&sup2;+b&sup2;-2ab)&sup2;&ge;0 (lu&ocirc;n đ&uacute;ng)}$      $ text{N&ecirc;n 2.(a&sup2;-ab+b&sup2;)&sup2;&ge;$a^{4}$ +$b^{4}$  (1)}$      $ text{Chứng minh tương tự}$      $ text{2.(b&sup2;-bc+c&sup2;)&sup2;&ge;$b^{4}$ +$c^{4}$  (2)}$      $ text{2.(c&sup2;-ac+a&sup2;)&sup2;&ge;$c^{4}$ +$a^{4}$ (3)}$      $ text{Nh&acirc;n (1);(2);(3) theo vế ta c&oacute;}$      $ text{8.(a&sup2;-ab+b&sup2;)&sup2;.(b&sup2;-bc+c&sup2;)&sup2;(c&sup2;-ac+a&sup2;)&sup2;&ge; ($a^{4}$ +$b^{4}$).($b^{4}$ +$c^{4}$).($c^{4}$ +$a^{4}$)=8}$      $ text{&hArr;(a&sup2;-ab+b&sup2;).(b&sup2;-bc+c&sup2;)(c&sup2;-ac+a&sup2;)&ge;1 }$

minhhaanh · Answer

Lời giải và giải thích chi tiết:   Ta sẽ đi chứng minh kết quả sau: $2.(a^2-ab+b^2)^2 &ge; a^4+b^4$   $&hArr;2.(a^4+b^4+a^2b^2+2a^2b^2-2ab(a^2+b^2)) &ge; a^4 +b^4$   $&hArr;(a^2+b^2-2ab)^2 &ge;&ge;0$   $&hArr;(a-b)^4 &ge;0$  (Lu&ocirc;n đ&uacute;ng)          (1)   Chứng minh ho&agrave;n to&agrave;n tương tự:   $&rArr;2.(b^2-bc+c^2)^2 &ge; b^4+c^4$ (2)   $&rArr;2.(c^2-ac+a^2)^2 &ge; c^4+a^4$  (3)   Nh&acirc;n c&aacute;c vế $(1).(2).(3)$:   $&rArr;8.(a^2-ab+b^2).(b^2-bc+c^2).(c^2-ac+a^2) &ge; (a^4+b^4).(b^4+c^4).(a^4+c^4)$   $&rArr;(a^2-ab+b^2).(b^2-bc+c^2).(c^2-ac+a^2) &ge;1$ (Đpcm)